: Voronoi diagrams are extremely versatile as a data
structure in many geometric applications. Computing this diagram ``exactly''
for a polyhedral set in 3-D has been a quest of computational geometers for
over two decades; this quest is still unrealized.We locate the difficulty in this quest, thanks
to a recent result of Everett et al (2009). More generally, it points to the
need for alternative computational models, and other notions of exactness. We
consider an alternative approach based on the well-known Subdivision Paradigm. A
brief review of such algorithms for Voronoi diagrams is given. Our unique
emphasis is the use of purely numerical primitives. We avoid exact (algebraic)
primitives because (1) they are hard to implement correctly, and (2) they fail
to take full advantage of subdivision.Our approach is captured by ``soft
primitives''
that conservatively converge to the exact ones in the limit.We illustrate our approach bydesigning the first purely numerical algorithmfor the Voronoi complex of a non-degenerate polygonal set.We also feature the critical role "Filters" in such algorithms.We will demo our preliminary implementations.Joint work with Vikram
Sharma (IMSc) and Jyh-Ming Lien (GMU)

- Lecture 2

✎Title : Pi = 3.14159... is in Log Space

✎Place : Math Science Building room 404

✎Date & Time : July 13, 2012 & 16:00 - 18:00

✎ Abstract

: A real number is in
"Log space" if its $n$-th bit ($n\in\ZZ$) can be computed in
logarithmic space (Log).
We show that Pi=3.14159... is in Log.This implies that Pi is in the complexity
class SC. The latter result was widely assumed to be true from famous BBP algorithm
for Pi by Bailey, Borwein and Plouffe. But Knuth, and independently Lipton,
pointed out that it is unclear that the BBP algorithm implies such a result. To
resolve this issue, we provide a new algorithm for Pi. Our result extends to
other constants such as log 2 or square of Pi that also possess two essential
ingredients: they have BBP-like series and have bounded irrationality
measures.

- Lecture 3

✎Title :Motion Planning and Theory of Soft Subdivision Search

✎Place : Math Science Building room 404

✎Date & Time : July 20, 2012 & 16:00 - 18:00

✎ Abstract

: We propose to design
motion planning algorithmsusing two ingredients:
the subdivision paradigm coupledwith ``soft predicates''.Such predicates are conservative and
convergent relative to traditional exact predicates. This leads to the concept
of ``resolution-exact'' algorithms. Resolution-exactness contains inherent
indeterminacies and other subtleties. We describe an algorithmic framework
called``Soft Subdivision Search'' (SSS) for designing such algorithms. There
are many parallels between our framework and the well-known Probabilistic Road
Maps (PRM) framework. Both frameworks lead to algorithms that are
highly practical, easy to implement, have adaptive and local complexity. The
critical difference is that SSS avoids the Halting Problem of PRM. We
demonstrate the ease of designing soft predicates for various motion planning
problems. The SSS framework provides a theoretically sound basis for new classes
of algorithms in motion planning and beyond. Such algorithms are novel, even in
the exact case. Our current implementation shows that such algorithms are not only
sound and simple to implement, but also efficient in practice. Joint Work with
Yi-Jen Chiang and Cong Wang.